Hexagonal Hierarchies and Close Packing of the Plane:
Overview
A scatter of points, spread evenly
across
the plane, may take on a variety of configurations: two simple
regular
lattices involve points that suggest squares or equilateral
triangles.
If one wishes to consider circular buffers around each point, then
these
buffers may overlap or be widely spaced. A natural issue to
consider is to provide some sort of maximal coverage of the plane by
the
buffers: to provide a "close packing" of the plane by circles.
Gauss
(1831/40) proved that the densest lattice packing of the plane is the
one
based on the triangular lattice. In 1968 (and earlier),
Fejes-Toth
proved that that same packing is not only the densest lattice packing
of
the plane but is also the densest of all possible plane packings.
If one thinks, then, of the circles as if they were bubble foam, the
circles
centered on a square grid pattern expand and collide to form a grid of
squares (Boys). The circles centered on a triangular grid pattern
expand and collide to form a mesh of regular hexagons, like the cells
in
a slice of the honeycomb of bees (de Vries). The theoretical
issues
surrounding tiling in the plane are complex; even deeper are those
issues
involving packings in three dimensional space. The reader
interested
in probing this topic further is referred to the Bibliography at the
end
of this document. Interpretation of the simple triangular grid
has
range sufficient to fill this document and far more.

Classical Urban Hexagonal Hierarchies
One classical interpretation of what dots
on a lattice might represent is found in the geometry of "central place
theory" (Christaller, Lösch). This idea takes the complex human
process of urbanization and attempts to look at it in an abstract
theoretical
form in order to uncover any principles which might endure despite
changes
over time, situation, cultural tradition, and all the various human
elements
that are truly the hallmarks of urbanization. Simplicity helps to
reveal form: models are not precise representations of
reality.
They do, however, offer a way to look at some structural elements of
complexity.
Thus, dots on a triangular lattice are populated places (often,
villages).
Circles, expanding into hexagons, are areas that are tributary to the
populated
places. In the traditional formulation (described after Kolars
and
Nystuen) one considers four basic postulates (no one of which is "real"
but each of which is simple):

The backdrop of land supports uniform population density

There is a maximum distance that residents can easily penetrate
into
the
tributary area.

There is slow, steady population growth

Village residents who move, as a result of growth (or for other
reasons),
attempt to remain in close contact with their previous location (to
maintain
social or other networks).

Suppose, in a triangular lattice of villages, that one village adds
to its retailing activities. After some time, growth occurs
elsewhere.
How might other villages compete to serve tributary areas: how
will
the larger, new villages share the tributary area? The answers
lead
to a surprising number of possible scenarios. Figure 1 shows the
first in an infinite number of possibilities. Animated locations,
for competing larger villages, are shown in Figure 1. The
smallest
villages are represented as small red dots; next nearest neighbors
competing
for intervening red dots are represented in blue; and, next nearest
neighbors
competing for intervening blue dots are represented in green. Of
course, one is usually only willing to travel so far to go to a place
only
slightly larger, so the fact that the animated pattern could be
extended
to an infinite number of levels, beyond green, may not mirror the
second
postulate. Over time, however, one might suppose further growth
and
an entire hierarchy of populated places.

Figure 1. A triangular
lattice of
dots with animated locations for competing locations entering and
vanishing
from the picture.

Virtual reality is an exciting form of
visualizing
three dimensional objects. Figure VR01 has a link on to a virtual
reality view of that figure: click on that figure to move into
the
virtual environment. Imagine the dots are holes in a pasta
machine
through which the pasta dough is to be extruded as spaghetti: the
view in Figure 1 is the template and the linked virtual reality is the
extruded pasta pulled through the red, blue, and green
holes.
Drive through this landscape; think of the view as a skyline of cell
towers
or some other tall thin structures (Arlinghaus, 1993). The
placement
of these towers is at vertices of equilateral triangles of various
sizes
forming an hexagonal hierarchy.

Figure VR01. A screen shot
from
the virtual world linked to this image. Click on the image to
enter
that world!

The pattern in Figure 1 suggests one
arrangement
for villages, towns, and cities. We offer systematic
visualization
schemes for a variety of such arrangements: first, following the
classical approaches to this issue found in the works of Walter
Christaller
and August Lösch and, second, following the contemporary approach
presented in previous conventional publications by the authors of this
submission.

CLASSICAL GEOMETRIC APPROACH TO HEXAGONAL HIERARCHIES

Visualization of Hexagonal Hierarchies using Animated Geometric
FiguresMarketing principle: K=3
Consider a central place point, A,
in a triangular lattice. Unit hexagons (fundamental cells)
surround
each of the points in the lattice and represent the small tributary
area
of each village (Figure 2). Growth at A has distinguished
it from other villages in the system. It will now serve a
tributary
area larger than will the unit hexagon. There are six villages directly
adjacent to A. The unit hexagons represent a partition of
area based on even sharing of area between A and these six
villages.
When A expands its central place activities, others may also
desire
to do so as well. Figure 2 shows the locations for the next
nearest
competitors to enter the system. Given that they, too, will share
area evenly, a set of larger hexagons emerges. Figure 3a shows
the
unit hexagons and the larger hexagons based on expansion of goods and
services.
The competitors that enter are spaced at a distance, in terms of
lattice
points spaced one unit apart, of
units (Figure 2). The position of the competitors that enter the
system in this scenario are as close as possible to A;
expansion
of goods and services at any of the six closest neighbors would
constitute
no change in basic pattern. One might imagine, therefore, that
emphasis
on distance minimization optimizes marketing capability--distance to
market
is at a minimum.

Figure 2. K=3: Marketing.Distance
measurement between adjacent competing new centers, A and A':
in this case, competing centers, blue dots, are spaced
units apart, assuming a distance of one unit between adjacent red dots.

Thus, when competitors are chosen in this manner, the pattern of one
layer of hexagons, in relation to another, has become known as a
hierarchy
arranged according to a "marketing principle" (Figure 3a).
Notationally,
it is captured by the square of the distance between competing
centers:
as a "K=3" hierarchy (Figure 2). Each large hexagon
contains
the equivalent of three smaller hexagons. One large hexagon = 1
small
hexagon + six copies of 1/3 of a small hexagon = 3 small hexagons
(Figure
3b, c, and d). Thus, the value K=3 is not only related to
distance between competing centers but also to size of tributary areas
generated by competition: as a constant of the hierarchy.

Figure 3a. K=3 hierarchy
showing
three layers of a nested hierarchy of hexagons of various sizes
oriented
with respect to one another according to the distance principle
illustrated
in Figure 2.

Figure 3b. Each
blue hexagon contains the equivalent of three red hexagons: one
entire
red hexagon surrounded by six copies of 1/3 of a red hexagon.

Figure 3c. Each green hexagon
contains
the equivalent of three blue hexagons: one entire blue hexagon
surrounded
by six copies of 1/3 of a blue hexagon.

Figure 4 shows the locations for the next nearest competitors, next
beyond those from K=3, to enter the system. Given that
they,
too, will share area evenly, a set of even larger hexagons
emerges.
Figure 5a shows the unit hexagons and the larger hexagons based on
expansion
of goods and services. The competitors that enter are
spaced
at a distance, in terms of lattice points spaced one unit apart, of 2
units
(Figure 4). The position of the competitors that enter the system
in this scenario lie along radials that fan outward from A and
pass
along existing boundaries to tributary areas. One might imagine,
therefore, that emphasis on market penetration, or transportation, is
the
focus here.

Figure 4. K=4: Transportation.Distance
measurement between adjacent competing new centers, A and A'
is 2 units, in this case (assuming a distance of 1 unit between
adjacent
red dots).

Thus, when competitors are chosen in this manner, the pattern of one
layer of hexagons, in relation to another, has become known as a
hierarchy
arranged according to a "transportation principle" (Figure 5a).
Notationally,
it is captured by the square of the distance between competing
centers:
as a "K=4" hierarchy (Figure 4). Each large hexagon
contains
the equivalent of four smaller hexagons. One large hexagon = 1
small
hexagon + six copies of 1/2 of a small hexagon = 4 small hexagons
(Figure
5b, c, d). Thus, the value K=4 is not only related to
distance
between competing centers but also to size of tributary areas generated
by competition--as a constant of the hierarchy.

Figure 5a. K= 4 hierarchy
showing
three layers of a nested hierarchy of hexagons of various sizes
oriented
with respect to one another according to the distance principle
illustrated
in Figure 4.

Figure 5b. Each
blue hexagon contains the equivalent of four red hexagons: one
entire
red hexagon surrounded by six copies of 1/2 of a red hexagon.

Figure 5c. Each green hexagon
contains
the equivalent of four blue hexagons: one entire blue hexagon
surrounded
by six copies of 1/2 of a blue hexagon.

Administrative principle: K=7
Figure 6 shows the locations for the next nearest competitors, next
beyond those from K=4, to enter the system. Given that
they,
too, will share area evenly, a set of even larger hexagons
emerges.
Figure 7a shows the unit hexagons and the larger hexagons based on
expansion
of goods and services. The competitors that enter are
spaced
at a distance, in terms of lattice points spaced one unit apart,
of
units (Figure 6). The position of the competitors that enter the
system in this scenario create larger hexagons whose boundaries pass
through
very few other populated places: hence, top-down control, or rule
from the center is emphasized. One might imagine, therefore, an
emphasis
on administrative control here.

Figure 6. K=7: Administrative.Distance
measurement between adjacent competing new centers, A and A'
is (assuming a
distance of
1 unit between adjacent red dots).

Thus, when competitors are chosen in this manner, the pattern of one
layer of hexagons, in relation to another, has become known as a
hierarchy
arranged according to an "administration principle" (Figure 7a).
Notationally, it is captured by the square of the distance between
competing
centers: as a "K=7" hierarchy (Figure 6). Each large
hexagon contains the equivalent of seven smaller hexagons. One
large
hexagon = 1 small hexagon + six copies of a small hexagon (underfit and
overfit regions balance) = 7 small hexagons (Figure 7b, c, d).
Thus,
the value K=7 is not only related to distance between competing
centers but also to size of tributary areas generated by
competition--as
a constant of the hierarchy.

Figure 7a. K= 7 hierarchy
showing
three layers of a nested hierarchy of hexagons of various sizes
oriented
with respect to one another according to the distance principle
illustrated
in Figure 6.

Figure 7b. Each blue hexagon
contains
the equivalent of seven red hexagons: one entire red hexagon
surrounded
by six copies equivalent to a single red hexagon. Each of the
perimeter
red hexagons is composed of 11/12 of a single red hexagonal cell plus
1/12
of an adjacent red cell: in an underfit/overfit pattern.

Figure 7c. Each green hexagon
contains
the equivalent of seven blue hexagons: one entire blue hexagon
surrounded
by six copies equivalent to a single blue hexagon. Each of the
perimeter
blue hexagons is composed of 11/12 of a single blue hexagonal cell plus
1/12 of an adjacent blue cell: in an underfit/overfit pattern.

One difficulty in constructing these geometric
visualizations
is that slight errors in placement of points get magnified in overlay
alignments.
To create a meshed hierarchy in which overlays are aligned is a
drafting
task of substantial proportions, when done by hand. Geographic
Information
System software, however, offers an easy and accurate method of
constructing
central place landscapes at almost any level of complexity (up to the
limits
of hardware and software capability). Figures 2-7 were created
using
ArcView GIS (v. 3.2, ESRI). The method for creating GIS-generated
central place landscapes employed the following steps:

obtain as a base map a triangular lattice shape file; such a file
may
be
created in ArcView using EdTools extension to precisely translate a
point.

ensure that each record in the underlying database has a unique
code
entered
in "number" format (using the "add record number" feature of Animal
Movement
extension, if need be).

if desired, create in a separate layer, a bounded region to serve
as
limits
within which to calculate the landscape--a rectangle, for
example.
One way to create such a region is to calculate the minimum convex
polygon
(convex hull) of the distribution of red dots using Home Range
extension.

load Spatial Analyst extension (ESRI) to ArcView and calculate
Thiessen
polygons using the Analysis|Assign Proximity command; choose the
rectangle
layer as the region within which to calculate the Thiessen
polygons.
Alternately, employ the same strategy using Home Range extension and
calculate
Dirichlet regions.

The result will appear as a set of small hexagons surrounding the
dots,
as in the red layers in Figures 2-7.

Repeat the procedure on other triangular lattices, with broader
spacing
of lattice points as in the blue and green points above, derived from
the
base lattice. The result will produce landscapes such as those in
Figures 2-7 depending on how the broader spacing pattern is selected.

The process of creating larger hexagons, as larger
tributary
areas representing expanded central place activities, can be carried
out
indefinitely. The set of figures above (3, 5, and 7) shows the
general
patterns that emerge and underscores, particularly, the importance of
the
constant of the hierarchy. Large hexagons in one layer contain
the
equivalent of K1 hexagons of the next smallest size
within
them; they contain the equivalent of K2 hexagons
from
the level two layers down in the hierarchy, and so forth. The K
value is an invariant of each geometric hierarchy that uniquely
characterizes
it. The mathematical search for invariants as bench marks against
which to view abstract structure is equivalent to the geographical
search
for bench marks in the field (physical or human) against which to view
mapped, spatial structure.

Visualization of Hexagonal Hierarchies using Mapplets
Another method of visualizing hexagonal hierarchies,
that is available only in current technology, looks simultaneously at
connection
patterns between multiple hierarchical layers of urban location maps
and
captures them as Java (TM) Applets: as "Mapplets." This
process
suggests a measure of visual stability of the geometric connectivity
pattern
that is related to the dimensions of the bounding box. Figures 8,
9, and 10 show Mapplets for the K=3, K=4, and K=7
hierarchies, respectively.

...ONLY in the original...see caveat at top of page...

Figure 8. K=3 Mapplet

Figure 9.K=4 Mapplet

Figure 10.K=7 Mapplet

Mapplets focus on connection patterns between successive
hierarchical
layers and, when K values are loaded as distances between
hierarchies,
they also suggest some elusive form of structural stability of
geometric
form. Animated maps of the central place geometry of the plane,
coupled
with mapplets showing animated hierarchical pattern alone, suggest a
three
dimensional view of central place geometry. A broader 3D view is
suggested in the next section.

CONTEMPORARY GEOMETRIC APPROACH TO HEXAGONAL HIERARCHIES

Visualization of Hexagonal Hierarchies using Animated Geometric
Figures
and Virtual Reality
In the material below, we illustrate the use
of the fractal concept of self-similarity to generate hexagonal
hierarchies
equivalent to those above, We use a hexagon as an initiator, and apply
to it different selections of generators, to produce the different
hexagonal
hierarchies of classical central place theory (based on original
concept
and work of S. Arlinghaus). In the previous sections we formed
central
place hexagonal hierarchies by moving from small hexagons to large
ones;
here, we reverse the process and dissect, using the self-similarity
transformation,
a large hexagon to create the smaller ones. In both processes,
the
results correspond exactly. The art is in generator selection,
and
it is simply that art that is presented in this chapter. Later
work
delves into the mathematical foundations of that art.

The K=3 Hierarchy When an hexagonal
initiator
is chosen and a two-sided generator, with included angle of 120
degrees,
is used to make successive replacement of the sides of the hexagon (as
in the animated Figure 11a), the outline of the next layer of the K=3
central place hierarchy is generated (the black lines in Figure 11a
suggest
interior connections). The replacement sequence applies the
generator
in an alternating pattern to the outside and then to the inside of the
initiator. When the original generator is scaled down, with shape
preserved, and applied in the outside/inside sequence to the newly
formed
blue polygon, the next lower level central place K=3 hierarchy
is
formed (as in the animated Figure 11b). The second, blue polygon
contains three scaled-down hexagons, self-similar to the first hexagon
(Figure 11a); the red polygon in the animation sequence contains three
shapes self-similar to the blue polygon (Figure 11b), and 27 (or 3
cubed)
hexagons self-similar to the original hexagon (Figure 11b). The
invariant
of 3, in the K=3 hierarchy, is replicated in this particular
fractal
iteration sequence.

It remains to determine
if the polygons generated in Figure 11 will in fact fit together to
form
the broad central place landscape of arbitrary size suggested in
Figure
3. To that end, we stack the layers generated above using the
fractal iteration sequence to form a tile of layers centered on the
single
polygonal initiator (Figure 12). Click here,
or on the screen shot in Figure VR02 below, to see a virtual solid
model
of the tile with which the reader can interact. Click
here,
or on the screen shot in Figure VR03 below, to see a virtual
translucent
model of the tile with which the reader can interact.

Figure VR02. A screen shot
from
the virtual world linked to this image. Click on the image to
enter
that world!

Figure VR03. A screen shot
from
the virtual world linked to this image. Click on the image to
enter
that world! Translucent solids permit one to see relationships
among
layers of the hexagonal hierarchy while travelling through the solids.

Figure 12. Fractally generated
layers
stacked on a single hexagonal tile. Click here
to see a virtual solid model of the tile with which the reader can
interact.
Click here to see a virtual
translucent
model of the tile with which the reader can interact.

Finally,
we tile the plane using the hexagonal initiators to discover if the
superimposed
structure also fits together perfectly (Figure 13). Hexagonal
tiles
are used to cover the plane without gaps, as is the case with the
sample
of green hexagons in Figure 13a. The hexagons mesh perfectly to
cover
the plane (Theorem of Gauss). In Figure 13b, the green outline of
the hexagons remains. Each of the solid green hexagons has had
the
fractal generator above applied and the consequent superimposed blue
tiles
come into view sequentially in this animation. Again, the fit is
exact, as we had hoped it might be. Finally, in Figure 13c, the
blue
outline only is retained from Figure 13b (along with the green outline
from Figure 13a). The final fractally generated layer derived
from
the blue polygons of Figure 13b comes into view in shades of red (or
yellow/gold
for contrast). The final layer of hexagonal base of unit hexagons
appears last. The fit is perfect: each green hexagon
contains
the equivalent of four blue hexagons and each blue hexagon contains the
equivalent of four red hexagons. The fractal generation procedure
created exactly the classical central place landscape of Figure
3. As the animation proceeds in Figure 13, further
layers
of the fractally generated hierarchy, attached to the tile in Figure
12,
come into view illustrating an exact meshing of tiles at all levels to
form a K=3 hierarchy.

Figure 13. Red layer on tile
from
Figure 12 fits exactly to form classical K=3 landscape.
Note,
that for contrast in blocks, the red layer from Figure 12 is
alternately
colored in shades of yellow, also.

The K=4 Hierarchy When an hexagonal initiator
is chosen and a three-sided generator, with included angles of 120
degrees
and shaped in the form of an isosceles trapezoid, is used to make
successive
replacement of the sides of the hexagon (as in the animated Figure
14a),
the outline of the next layer of the K=4 central place
hierarchy
is generated (the black lines in Figure 14a suggest interior
connections).
The replacement sequence applies the generator in an alternating
pattern
to the outside and then to the inside of the initiator. When the
original generator is scaled down, with shape preserved, and applied in
the outside/inside sequence to the newly formed blue polygon, the next
lower level central place K=4 hierarchy is formed (as in the
animated
Figure 14b). The second, blue polygon contains four scaled-down
hexagons,
self-similar to the first hexagon (Figure 14a); the red polygon in the
animation sequence contains four shapes self-similar to the blue
polygon
(Figure 14b), and 64 (or 4 cubed) hexagons self-similar to the original
hexagon (Figure 14b). The invariant of 4, in the K=4
hierarchy,
is replicated in this particular fractal iteration sequence.

It remains to determine
if the polygons generated in Figure 14 will in fact fit together to
form
the broad central place landscape of arbitrary size suggested in
Figure
5. To that end, we stack the layers generated above using the
fractal iteration sequence to form a tile of layers centered on the
single
polygonal initiator (Figure 15). Click
here,
or on the screen shot in Figure VR04 below, to see a virtual solid
model
of the tile with which the reader can interact. Click
here,
or on the screen shot in Figure VR05 below, to see a virtual
translucent
model of the tile with which the reader can interact.

Figure VR04. A screen shot
from
the virtual world linked to this image. Click on the image to
enter
that world!

Figure VR05. A screen shot
from
the virtual world linked to this image. Click on the image to
enter
that world! Translucent solids permit one to see relationships
among
layers of the hexagonal hierarchy while travelling through the solids.

Figure 15. Fractally generated
layers
stacked on a single hexagonal tile. Click here
to see a virtual solid model of the tile with which the reader can
interact.
Click here to see a virtual
translucent
model of the tile with which the reader can interact.

Finally,
we tile the plane using the hexagonal initiators to discover if the
superimposed
structure also fits together perfectly (Figure 16). Hexagonal
tiles
are used to cover the plane without gaps, as is the case with the
sample
of green hexagons in Figure 16a. The hexagons mesh perfectly to
cover
the plane (Theorem of Gauss). In Figure 16b, the green outline of
the hexagons remains. Each of the solid green hexagons has had
the
fractal generator above applied and the consequent superimposed blue
tiles
come into view sequentially in this animation. Again, the fit is
exact, as we had hoped it might be. Finally, in Figure 16c, the
blue
outline only is retained from Figure 16b (along with the green outline
from Figure 16a). The final fractally generated layer derived
from
the blue polygons of Figure 16b comes into view in shades of red (or
yellow/gold
for contrast). The final layer of hexagonal base of unit hexagons
appears last. The fit is perfect: each green hexagon
contains
the equivalent of four blue hexagons and each blue hexagon contains the
equivalent of four red hexagons. The fractal generation procedure
created exactly the classical central place landscape of Figure
5. As the animation proceeds in Figure 16, further
layers
of the fractally generated hierarchy, attached to the tile in Figure
15,
come into view illustrating an exact meshing of tiles at all levels to
form a K=4 hierarchy.

Figure 16c. Red layer on tile
from
Figure 15 fits exactly to form classical K=4 landscape.
Note,
that for contrast in blocks, the red layer from Figure 15 is
alternately
colored in shades of yellow, also.

The K=7 Hierarchy When an hexagonal
initiator
is chosen and a three-sided generator, with included angles of 120
degrees
and shaped in a zig-zag form, is used to make successive replacement of
the sides of the hexagon (as in the animated Figure 17a), the outline
of
the next layer of the K=7 central place hierarchy is generated
(the
black lines in Figure 17a suggest interior connections). The
replacement
sequence applies the generator in an alternating pattern to the outside
and then to the inside of the initiator. When the original
generator
is scaled down, with shape preserved, and applied in the outside/inside
sequence to the newly formed blue polygon, the next lower level central
place K=7 hierarchy is formed (as in the animated Figure
17b).
The second, blue polygon contains seven scaled-down hexagons,
self-similar
to the first hexagon (Figure 17a); the red polygon in the animation
sequence
contains seven shapes self-similar to the blue polygon (Figure 17b),
and
343 (or 7 cubed) hexagons self-similar to the original hexagon (Figure
17b). The invariant of 7, in the K=7 hierarchy, is
replicated
in this particular fractal iteration sequence..

It remains to determine
if the polygons generated in Figure 17 will in fact fit together to
form
the broad central place landscape of arbitrary size suggested in
Figure
7. To that end, we stack the layers generated above using the
fractal iteration sequence to form a tile of layers centered on the
single
polygonal initiator (Figure 18). Click
here,
or on the screen shot in Figure VR06 below, to see a virtual solid
model
of the tile with which the reader can interact. Click
here,
or on the screen shot in Figure VR07 below, to see a virtual
translucent
model of the tile with which the reader can interact.

Figure VR06. A screen shot
from
the virtual world linked to this image. Click on the image to
enter
that world!

Figure VR07. A screen shot
from
the virtual world linked to this image. Translucent solids permit
one to see relationships among layers of the hexagonal hierarchy while
travelling through the solids. Click on the image to enter that
world:
blast off in this virtual hexagonal space ship!

Figure 18. Fractally generated
layers
stacked on a single hexagonal tile. Click here
to see a virtual solid model of the tile with which the reader can
interact.
Click here to see a virtual translucent
model of the tile with which the reader can interact.

Finally,
we tile the plane using the hexagonal initiators to discover if the
superimposed
structure also fits together perfectly (Figure 19). Hexagonal
tiles
are used to cover the plane without gaps, as is the case with the
sample
of green hexagons in Figure 19a. The hexagons mesh perfectly to
cover
the plane (Theorem of Gauss). In Figure 19b, the green outline of
the hexagons remains. Each of the solid green hexagons has had
the
fractal generator above applied and the consequent superimposed blue
tiles
come into view sequentially in this animation. Again, the fit is
exact, as we had hoped it might be. Finally, in Figure 19c, the
blue
outline only is retained from Figure 19b (along with the green outline
from Figure 19a). The final fractally generated layer derived
from
the blue polygons of Figure 19b comes into view in shades of red (or
yellow/gold
for contrast). The final layer of hexagonal base of unit hexagons
appears last. The fit is perfect: each green hexagon
contains
the equivalent of four blue hexagons and each blue hexagon contains the
equivalent of four red hexagons. The fractal generation procedure
created exactly the classical central place landscape of Figure
7. As the animation proceeds in Figure 19, further
layers
of the fractally generated hierarchy, attached to the tile in Figure
18,
come into view illustrating an exact meshing of tiles at all levels to
form a K=7 hierarchy.

Figure 19c. Red layer on tile
from
Figure 18 fits exactly to form classical K=7 landscape.
Note,
that for contrast in blocks, the red layer from Figure 18 is
alternately
colored in shades of yellow, also.

Thus, the complex mechanics of classical central place theory come
alive
as a single dynamic system when viewed using fractal geometry.
The
fit is exact.

The Added Role of the Fractional DimensionA fractal iteration sequence, such
as those above but carried out infinitely, might be thought to increase
the extent to which a line "fills" space. Both a single line
segment
and the letter "N" have Euclidean dimension 1; yet one of them fills
more
space than does the other. Mandelbrot (and others before him)
captures
this notion of space-filling with the concept of fractional dimension
(hence
"fractal"). He uses Hausdorff-Besicovitch dimension to measure
the
enduring mathematical concept of space-filling. We employ
Mandelbrot's
formulation for fractional dimension D as, D=log(number
of
generator sides)/log(square root of K). Thus, the
following
values for fractally-generated central place hierarchies emerge:

K=3, D=log2/log =
1.2618595

K=4, D=log3/log 2 = 1.5849625

K=7, D=log3/log =
1.1291501

The idea with the space-filling is to pick an
arbitrary
point in the bounded space containing the curve. Place a circle
of
arbitrarily small radius around that point. Does that circle
contain
a point on the curve as the fractal iteration sequence goes to
infinity?
If that is the case for any point, then the curve is said to fill space
and have dimension 2. If not, then there are holes or gaps
(perhaps
of infinitesimal size) in the space and the curve fails to fill space
completely
and has fractional dimension between 1 and 2 (as a sort of Swiss
cheese,
Emmenthaler, with holes). Thus, the K=4 fractal iteration
sequence, if permitted to repeat infinitely, has the highest fractional
dimension of these three: this curve gets "closer" to arbitrary
points
in space than do the lines of the other hierarchies, as one might hope
a hierarchy interpreted as a "transportation" hierarchy would.
The
fractional dimension of the fractal iteration sequence corresponds to
the
intuitive notion of scholars over time as to interpretation: as
another
benchmark or field test of theory. The K=7 fractal
iteration
sequence, if permitted to repeat infinitely, has the lowest fractional
dimension of these three, keeping control from the center optimized and
hence supporting the "administrative" interpretation often given to the
classical K=7 hierarchy. Finally, the K=3 falls
between:
marketing needs greater spatial penetration than does administration
but
less than does transportation. Here, the fit between classical
interpretation
and fractal calculation is reasonable (one could never say "exact"
because
the terms "marketing," "transportation," and "administrative" are
inexact
terms themselves).

What is difficult with fractals is to visualize
the
infinite process. Graphic color display, including three
dimensional
display, offers exciting strategies for visualization. Very
quickly,
however, it becomes difficult to draw the fine lines required by
repeating
the process at more and more local scales: lines have
width.
Electronic lines can be controlled and made finer than can pen lines,
but
eventually the line-width limits the capability to produce graphic
images.
Eventually, the mind's eye must take over and extrapolate the visual
infinite
process.
Another possibility might be to draw on the other
human senses to aid in that extrapolation. Thus, Figure 20 shows
figures generated by Fractal Music 1.9; click on the images and hear
the
associated music. The left figure shows the cellular automata base
generated
by default--it is bilaterally symmetric about a central vertical line
and
was generated using a symmetrically arranged initiator string of 64
digits
ranging in value from 0 to 7 (one for each tone). The next
figure,
K=3,
shows the cellular automata diagram (another sort of "bubble foam" in
appearance)
generated using the value for the fractional dimension of the K=3
hexagonal hierarchy carried out to 64 decimal places as the initiator
string
for the music. The next figure, K=4, shows the cellular
automata
diagram generated using the value for the fractional dimension of the K=4
hexagonal hierarchy carried out to 64 decimal places as the initiator
string
for the music. The final figure, K=7, shows the cellular
automata
diagram generated using the value for the fractional dimension of the K=7
hexagonal hierarchy carried out to 64 decimal places as the initiator
string
for the music. Click on each figure to hear the music. Each
musical sequence, of over 1000 steps, was created from the default
base,
changing only the initiator string, so that the fractional dimension is
what operates on a "seed" value of basic notes. The listener
should
hear the basic pattern in all characterizations: great symmetry
in
the base value; abrupt changes of state in the K=3 value; a
smoother
filling of musical space in the K=4 music; and, gaps in the K=7
musical characterization derived from the K=7 fractional
dimension.
Thus, we extend visualization from two dimensional graphical images to
three dimensional graphical images to the mind's eye, and finally, to
the
mind's ear: capturing hierarchical pattern through 1000 steps or
more is easy in the musical clips. Such characterization offers added
capability
to those of us with all of our senses that are functional: for
those
with limited visual sensory function, it offers a way to an auditory
"visualization"
of the beauty of geometry.

Base

K=3

K=4

K=7

Figure 20. Fractal music connection. Click on the
images
generated by the fractional dimensions of the hexagonal hierarchies;
compare
these to the default base created by the software.
Future Directions

The complex mechanics of the theory behind
hexagonal
hierarchies come alive as a single dynamic system when visualized
through
the lens of fractal geometry. The fit of the classical and
fractal
geometric hierarchies is exact. Thus, as one might use a
carefully
surveyed topographic map, with field-checked spot elevations, as a
guide
into dense jungle or other unsurveyed landscapes, so too we use our
carefully
surveyed alignment of the classical and the fractal hexagonal
hierarchies
as a guide into unseen or unproven areas of geometry and
geography.
The difference is that the "field" tests in one case occur "terrestrial
space" while in the other the "field" tests occur in "geometry, number
theory, and pure mathematics."

In the material above, we saw hexagonal hierarchies, of different
orientation,
cell size, and stacking characteristics, arise from the same base of
unit
hexagons. These were associated with three integers: 3, 4,
and 7. The thoughtful reader might naturally ask a number of
questions,
such as:

are there other numbers that would serve as K values or
are 3,
4,
and 7 the only such values?

are 5 or 6 possible K values?

are there K values larger than 7?

how many K values are there?

How does one determine the number of sides in a fractal generator
that
will generate a correct hierarchy for arbitrary K values?

How does one determine fractal generator shape that will generate
a
correct
hierarchy for arbitrary K values?

Earlier research, by August Lösch, Michael Dacey,
and others shows illustrations of K-values greater than
7.
Indeed, research by Arthur Loeb, in crystallography, and Dacey, in
geography,
led to independent discovery that the Diophantine equation, x2+xy+y2
would generate all K values when pairs of positive integers
were
substituted for x and for y. Thus, when (x,y)=(1,1)
the equation x2+xy+y2 = K
yields
a value of K=3; when (x,y)=(0,2), it follows that
K=4;
and, when (x,y)=(1,2), it follows that K=7.
Pairs such as (0,0) and (1,0) yield only trivial results so that the
values
of 3, 4, and 7 are the three smallest K-values. There are
no other K values less than 7.

The result of Loeb/Dacey is important because it shows

that there are an infinite number of possible K values

that this infinity of values is in one-to-one correspondence with
the
integral
lattice points in the plane

that one can give a numerical generating function to create K
values

Thus, a graph of lattice points in the plane offers a convenient method
of visualizing K-values (Figure 21). The animation shows
the
coordinate pairs in this oblique coordinate system with axes inclined
at
60 degrees (instead of the conventional 90 degrees). The
coordinate
pairs are replaced in animated fashion by single numbers representing
the
K
value that corresponds to that ordered pair.

Figure 21. The coordinatized
lattice
points in yellow transform into K-values in cyan using the
Diophantine
equation K=x2+xy+y2

Previous published research by the authors of this presentation has
shown how to determine the number of sides in a fractal generator that
will generate a correct hierarchy for arbitrary K values and
how to determine fractal generator shape that will generate a correct
hierarchy for arbitrary K values. Work in progress shows
how
to extend the three dimensional and other visualization schemes shown
here
to higher K values. In it, we offer mathematical proof of
these ideas and extensions of them into new realms. The classical
is used for alignment of new with the old: a strategy useful in a
wide range of theoretical and applied research.